====== Estrous 2 Genders ======

\( \require{newcommand} \)
\( \newcommand{\diff}[2]{\frac{d{#1}}{d{#2}}} \)
\( \newcommand{\abs}[1]{\left|#1\right|} \)

Estrous with two genders. This is an extension of [[Estrous 1 Gender (Redux)]]. Now we have anestrous and estrous females (A and E) and males (M).  N represents any individual (A, E, or M).

===== Processes =====

\[
  A \xrightarrow{\alpha} E 
\]
\[
  E \xrightarrow{\gamma} A 
\]
\[
  M + E \xrightarrow{\beta/2} M + 2 A 
\]
\[
  M + E \xrightarrow{\beta/2} 2 M + A 
\]
\[
  2 N \xrightarrow{\delta} N 
\]

The first two are strictly physiological processes (transitions between anestrous and estrous physiological states) while the last three are ecological (affecting the population density).

===== Full Dynamics =====

\[
  \diff{a}{t} = - \alpha a + \gamma e + \frac{3}{2} \beta m e - \delta a n 
\]
\[
  \diff{e}{t} = + \alpha a - \gamma e - \beta m e - \delta e n 
\]
\[
  \diff{m}{t} = + \frac{1}{2} \beta m e - \delta m n 
\]
\[
  \diff{n}{t} = + \beta m e - \delta n^2 
\]

where \(n=a+e+m\).

===== Gender Balance =====

To simplify the subsequent analysis let's look at the gender dynamics:

\[
  \diff{f}{t} = \diff{a}{t} + \diff{e}{t} = + \frac{1}{2} \beta me - \delta f n 
\]
\[
  \diff{m}{t} = + \frac{1}{2} \beta m e - \delta m n 
\]

where \(f = a+e \) is the female density.

Then the dynamics of the gender imbalance, \( g=f-m\) follow 
\[
  \diff{g}{t} = - \delta g n 
\]
so gender-indiscriminate competition promotes balance of the gender densities, \(g \rightarrow 0\).

For convenience the following analysis will assume that any initial imbalance has been eliminated and the genders are equally represented:
\[
  f = m = n / 2. 
\]

Note: we will be assuming the gender balance timescale, \( 1 / \delta n \), is a long time compared to other dynamics in our system.  Nevertheless, we may be able to justify the gender balance initial condition by further assuming that the lineage has been converging to gender balance over evolutionary time prior to our involvement.

===== Slow Timescale =====

==== Units ====

Let's assume that ecological processes are much slower than physiological ones.  In particular, let's assume

\[
  \beta \abs{K}, \delta \abs{K} \ll \alpha, \gamma 
\]

where \( \abs{K} \) is a population density constant, ie. it has the same units as \(n\), \( [K] = [n] \) (where \( [\cdot] \) means "units of").  We choose \( \abs{K} \) so it is a characteristic population size for the dynamics.  That way we can compare the rate constants, because \( K \) matches the units: 

\[
  [\beta K] = [\delta K] = [\alpha] = [\gamma]. 
\]

==== Separation of Timescales ====

We can now specify what we mean when we say "ecological processes are much slower than physiological ones".  Let us define \( \hat{\beta} \) and \( \hat{\delta} \) so that

\[
  \beta \equiv \epsilon \hat{\beta} / \abs{K}, 
\]
\[
  \delta \equiv \epsilon \hat{\delta} / \abs{K} 
\]

where \(\epsilon \ll 1 \) is a dimensionless constant.  Since \(\epsilon \ll 1\) the ecological processes (involving \(\beta\) and \(\delta\)) occur much less frequently than the physiological transitions (with \(\alpha\) and \(\gamma\)).

Now we can look at the dynamics under our original (fast) time variable \(t\) or with a new, slow, time variable, \( \hat{t} \equiv \epsilon t \).  In this slower timescale we find the dynamics become

\[
  \epsilon a' \equiv \epsilon \diff{a}{\hat{t}} = - \alpha a + \gamma e + \frac{3}{2} \epsilon \hat{\beta} m e / \abs{K} - \epsilon \hat{\delta} a n / \abs{K} 
\]
\[
  \epsilon e' \equiv \epsilon \diff{e}{\hat{t}} = + \alpha a - \gamma e - \epsilon \hat{\beta} m e / \abs{K} - \epsilon \hat{\delta} e n / \abs{K} 
\]
\[
  m'          \equiv \diff{m}{\hat{t}} = + \frac{1}{2} \hat{\beta} m e / \abs{K} - \hat{\delta} m n / \abs{K} 
\]
\[
  n'          \equiv \diff{n}{\hat{t}} = + \hat{\beta} m e / \abs{K} - \hat{\delta} n^2 / \abs{K}. 
\]

where the prime, \( \cdot' \), denotes the slow derivative, \( \diff{\cdot}{\hat{t}} \).

Note that \(\epsilon\) cancels out in the dynamical equations for \(m\) and \(n\) which means that \(m\) and \(n\) variy at a measurable rate on the slow timescale--\(m\) and \(n\) are //slow// variables.

===== Quasi-Steady-State Assumption =====

So far our analysis has been exact.  Now let's assume that the other, //fast// variables equilibrate quickly so that \( \epsilon a' \approx 0 \approx \epsilon e' \), the quasi-steady-state assumption, QSSA.  Then we can compute the quasi-steady-state densities \(a\) and \(e\) at any time in terms of the slow variables:

\[
  a \approx \frac{2 \gamma \abs{K} + 3 \epsilon \hat{\beta} m}{2 (\alpha + \gamma) \abs{K} + \epsilon (3 \hat{\beta} m + 2 \hat{\delta} n)} f 
\]
\[
  e \approx \frac{\alpha \abs{K}}{(\alpha + \gamma) \abs{K} + \epsilon (\hat{\beta} m + \hat{\delta} n)} f. 
\]

Fortunately, this gives us a way to check the validity of our assumption: we originally required \(a+e+m=n\) so this should still be largely correct.  Taking a Taylor expansion around \(\epsilon=0\) gives a first-order approximation (with gender balance):

\[
  a+e+m \approx n \left[ 1 + \epsilon \frac{ \alpha\hat{\beta} - 4 (\alpha + \gamma) \hat{\delta}) }{8 (\alpha+\gamma)^2 \abs{K}} n + \ldots \right] \equiv n \left[ 1 \pm n/n_\epsilon + \ldots \right]. 
\]

This tells us the QSSA is only valid when

\[
  n \ll n_\epsilon = \frac{1}{\epsilon} \frac{8 (\alpha+\gamma)^2 \abs{K}}{ \abs{\alpha \hat{\beta} - 4 (\alpha + \gamma) \hat{\delta})} } = \frac{8 (\alpha+\gamma)^2}{\abs{\alpha \beta-4 (\alpha + \gamma) \delta}}. 
\]

==== Alternate QSSA Condition ====

Originally, we assumed \( \delta \abs{K} \ll \alpha \) (among other things).  That puts a condition on \(\abs{K}\) for QSSA to be satisfied:

\[
  \abs{K} \ll Q \equiv \frac{\alpha}{\delta}. 
\]

===== QSSA Birthrate =====

Recall, the population dynamics on the slow, ecological timescale are

\[
  n' = + \hat{\beta} m e / \abs{K} - \hat{\delta} n^2 / \abs{K}. 
\]

The variables \(m\) and \(e\) can be written in terms of the total population density, \(n\), under QSSA (and gender balance):

\[
  m = n/2 
\]
\[
  e \approx \frac{\alpha \abs{K}}{2 (\alpha + \gamma) \abs{K} + \epsilon (\hat{\beta} + 2 \hat{\delta}) n } n. 
\]

So, when QSSA holds we can approximate the population dynamics on the slow timescale by

\[
  n' \approx \frac{\alpha \hat{\beta}}{(\alpha + \gamma) \abs{K}} \frac{n}{1+n/n_\mathrm{Allee}} n - \hat{\delta} n^2 / \abs{K}. 
\]

where
\[
  n_\mathrm{Allee} \equiv \frac{2 (\alpha + \gamma)}{\epsilon (\hat{\beta} + 2 \hat{\delta})} \abs{K} = \frac{2 (\alpha+\gamma) }{\beta + 2 \delta}.
\]

<div right round box>
{{estrous2gpercapitabirthrate.png?300}}

Figure 1: Per-capita birthrate, \(B(n)\) for population dynamics of \(n\) given QSSA assumption.
</div>

On the fast timescale this becomes

\[
  \diff{n}{t} \approx B(n) n - \delta n^2 
\]

where the per-capita birthrate, \(B(n)\), is
\[
  B(n) \equiv \frac{\alpha \beta}{4(\alpha + \gamma)} \frac{n}{1 + n/n_\mathrm{Allee}} = r \frac{n}{n_\mathrm{Allee} + n} 
\]
and the maximum per-capita birthrate, \(r\), is
\[
  r \equiv n_\mathrm{Allee} \frac{\alpha \beta}{4(\alpha + \gamma)} = \frac{\alpha \beta}{2(\beta + 2 \delta)}. 
\]

==== Logistic domain ====

If \(n_\mathrm{Allee} \ll n \ll n_\epsilon\) then the dynamics lie within the //logistic domain// where the per-capita birthrate is roughly constant, \(B(n)\approx r\), and the population dynamics follow

\[
  \diff{n}{t} \approx r n - \delta n^2, 
\]

the logistic rate equation.

==== Allee domain ====

On the other hand, if the population is small, \( n \ll n_\mathrm{Allee} \), then the birthrate is suppressed, the Allee effect.  In this domain the per-capita birthrate is roughly \(B(n)\approx r n / n_\mathrm{Allee} \) and the population follows

\[
  \diff{n}{t} \approx \left( \frac{r}{n_\mathrm{Allee}} - \delta \right) n^2. 
\]

Clearly, the population is unstable if \( r < \delta n_\mathrm{Allee} \).  This would reflect a //strong// Allee effect, where the mating rate drops to an unsupportable level if the population is too small.  On the other hand, if \( r > \delta n_\mathrm{Allee} \) (or \( \alpha \beta > \delta (\alpha+\gamma) \)) then we have a //weak// Allee effect--the birthrate is suppressed compared to the logistic equation but the population will nevertheless grow if undisturbed.

===== Carrying capacity =====

Regardless of whether QSSA holds we can find the non-trivial equilibrium population density. Since this represents the //characteristic// population size we can choose this to represent our parameter \(K\) from above:

\[
  K \equiv \frac{r}{\delta} - n_\mathrm{Allee} = \frac{\alpha \beta - 4 (\alpha + \gamma) \delta }{2 (\beta + 2 \delta) \delta}. 
\]

(The first equality is expressed in terms of the above QSSA analysis but the second equality represents the equilibrium \(n\)-density even if QSSA is violated.)

I haven't done the Jacobian stability analysis for the full system but it looks pretty clear that \(n=K\) is only stable when \(K>0\), otherwise \(n=0\) is stable.  Then if \(K\leq 0\) the population is unstable and will inevitably go extinct.  So we only expect \(K>0\) to represent naturally-occurring populations (unless the parameters have changed recently to destabilize the population).

If \(K>0\) (\( \alpha \beta > 4 (\alpha + \gamma) \delta \)) then the //weak// Allee effect applies in the Allee domain (\( \alpha \beta > (\alpha + \gamma) \delta\)) so the population is globally stable (regardless of how small it gets).  On the other hand, if \(K<0\) then the system is unstable for all \(n\).

One final point about the carrying capacity.  For comparison with the logistic equation, in terms of \(r\), \(K\), and \(n_\mathrm{Allee}\) the QSSA dynamics can be written

\[
  \diff{n}{t} = r n \left( \frac{n}{n+n_\mathrm{Allee}} - \frac{n}{K+n_\mathrm{Allee}} \right). 
\]

===== Parameter space =====

The possible dynamics can be expressed more concisely by reducing the parameter space to two dimensions: 

\[
  x=\gamma/\alpha 
\]
\[
  y=\beta/\delta. 
\]

We will find there are five qualitatively separate regions in this parameter space, depending on the relationships between the constants \(n_\epsilon, K\), \(n_\mathrm{Allee}\), and \(Q\).  In terms of \(x\) and \(y\) these constants can be written as

\[
  n_\epsilon = \frac{\alpha}{\delta} \frac{8 (1+x)^2}{ \abs{y - 4 (1+x)} } 
\]
\[
  K = \frac{\alpha}{\delta} \frac{y - 4 (1+x)}{2 (y+2)} 
\]
\[
  n_\mathrm{Allee} = \frac{\alpha}{\delta} \frac{2(1+x)}{y+2} 
\]
\[
  Q = \frac{\alpha}{\delta}. 
\]

==== Stability ====

If \(K\leq 0\) then we expect the population to be unstable and to crash (under both the full dynamics and the QSSA approximation).  In terms of \((x,y)\) this is

\[
  \mathrm{Unstable} : y\leq 4 (1+x). 
\]

==== QSSA limit ====

If \( K > 0\)  (\(y>4(1+x)\)) then \(K\) represents a characteristic population density (actually, the equilibrium) so it is reasonable to expect the actual density to be on the same order, \( n \sim K \).  Then we can express the QSSA condition as \( K \ll n_\epsilon \).  This condition implies

\[
  \frac{y - 4 (1+x)}{2 (y+2)} \ll \frac{8 (1+x)^2}{ y - 4 (1+x) } 
\]

or

\[
  \mathrm{QSSA} : y \ll 4 \left( 3+5x+2x^2 + \sqrt{2} \sqrt{ 5+16x+19x^2+10x^3+2x^4 } \right) . 
\]

=== Alternative QSSA limit ===

\(\abs{K}\) represents the characteristic population density.  A QSSA condition was \( \abs{K} \ll Q = \alpha / \delta \), so in terms of \((x,y)\) this becomes

\[
  y - 4 (1+x) \ll 2 (y+2) \textrm{ if } K\geq 0 
\]
\[
  4 (1+x) - y \ll 2 (y+2) \textrm{ if } K<0. 
\]

The first form is approximately valid for all \((x>0, y>4(1+x))\).  The second form, which occurs in the unstable regime, implies

\[
  \textrm{unstable QSSA} : y \gg \frac{4}{3} x. 
\]

==== Logistic vs Allee domain ====

If \(K>0\) and the QSSA applies then the dynamics will carry \(n\) towards \(K\).  If \(K<n_\mathrm{Allee}\) then the population will always be in the Allee domain with a depressed population growth rate.  This condition is simply

\[
  \mathrm{Allee} : y < 8(1+x). 
\]

==== Summary ====

<div right round box>
{{:research:draft:2009:estrous2gparameterspace.png?300}}

Figure 2: Parameter space can be divided into five regions characterizing the dynamics.  The red dots indicates points where the dynamics are computed numerically (see below).
</div>

In conclusion, there are five regions of interest in the \((x,y)\)-plane:

\[
  \textrm{stable non-QSSA} : Q > K \gtrsim n_\epsilon > n_\mathrm{Allee} > 0 
\]
\[
  \textrm{stable logistic QSSA} : (Q, n_\epsilon) \gg K > n_\mathrm{Allee} > 0 
\]
\[
  \textrm{stable Allee QSSA} : n_\epsilon > Q \gg n_\mathrm{Allee} > K > 0 
\]
\[
  \textrm{unstable QSSA} : n_\epsilon > (Q, n_\mathrm{Allee}) \gg \abs{K} > 0 > K 
\]
\[
  \textrm{unstable non-QSSA} : n_\epsilon > n_\mathrm{Allee} > \abs{K} > Q > 0 > K 
\]

Note that \(n_\epsilon > n_\mathrm{Allee}\) always.

===== Numerics =====

To test our analytic results we can construct a numeric simulation.  The above analysis suggests the regions we should check numerically (marked in red on Figure 2).  The results, shown below, are better than could be expected.  The QSSA tracks the full dynamics quite well in all cases.

The numerics show QSSA to be a more robust approximation than would be expected from the conditions \(n\ll Q\) or \(n\ll n_\epsilon\).  

| Region   | \( x = 0 \) | \( x = 1 \) | \( x = 4 \) | \( x = 16 \) |
| Stable non-QSSA \\ \( Q > K > n_\epsilon > n_{Allee} \) |  {{estrous2g20091023_stablenon_x0y37.2982.png?150}} |  {{estrous2g20091023_stablenon_x1y121.5843.png?150}} |  {{estrous2g20091023_stablenon_x4y661.8144.png?150}} |  {{estrous2g20091023_stablenon_x16y7141.9425.png?150}} |
| Logistic \\ \( (Q, n_\epsilon) > K > n_{Allee} \) |  {{estrous2g20091023_logistic_x0y12.png?150}} |  {{estrous2g20091023_logistic_x1y40.png?150}} |  {{estrous2g20091023_logistic_x4y220.png?150}} |  {{estrous2g20091023_logistic_x16y2380.png?150}} |
| Allee \\ \( n_\epsilon > Q > n_{Allee} > K \) |  {{estrous2g20091023_allee_x0y6.png?150}} |  {{estrous2g20091023_allee_x1y12.png?150}} |  {{estrous2g20091023_allee_x4y30.png?150}} |  {{estrous2g20091023_allee_x16y102.png?150}} |
| Unstable QSSA\\ \( n_\epsilon > (Q, n_{Allee}) > \abs{K} \)\\ \( 0 > K \) |  {{estrous2g20091023_unstableqssa_x0y0.png?150}} |  {{estrous2g20091023_unstableqssa_x1y2.png?150}} |  {{estrous2g20091023_unstableqssa_x4y8.png?150}} | {{estrous2g20091023_unstableqssa_x16y32.png?150}} |
| Unstable non-QSSA\\ \( n_\epsilon > n_{Allee} > \abs{K} > Q \)\\ \( 0 > K \) |  {{estrous2g20091023_unstablenon_x0y0.png?150}} |  {{estrous2g20091023_unstablenon_x1y0.66667.png?150}} |  {{estrous2g20091023_unstablenon_x4y2.6667.png?150}} |  {{estrous2g20091023_unstablenon_x16y10.6667.png?150}} |

===== Comparison to 1 Gender =====

In [[Estrous 1 Gender (Redux)]] we developed similar arguments for a simpler model with sexual reproduction amongst a single gender.  The conclusions were qualitatively the same but differ in detail.  Both place QSSA conditions on the population density, \( n \ll Q, n_\epsilon \), in which case the dynamics follow

\[
  \diff{n}{t} = r n \left( \frac{n}{n+n_\mathrm{Allee}} - \frac{n}{K+n_\mathrm{Allee}} \right). 
\]

The parameters have similar forms.

| Parameter  |  1 Gender  |  2 Genders  |
| \( n_\epsilon \)\\ QSSA condition, \( n \ll n_\epsilon \)  |  \[  \frac{(\alpha+\gamma)^2}{\abs{\alpha \beta - (\alpha + \gamma) \delta}} \]  |  \[  \frac{8 (\alpha+\gamma)^2}{\abs{\alpha \beta-4 (\alpha + \gamma) \delta}} \]  |
| \( Q \)\\ QSSA condition, \( n \ll Q \)  |  \[  \alpha / \delta \]  |  \[  \alpha / \delta \]  |
| \( n_\mathrm{Allee} \)\\ Allee condition, \( n \ll n_\mathrm{Allee} \)  |  \[  \frac{\alpha+\gamma}{\beta+\delta} \]  |  \[  \frac{2 (\alpha+\gamma) }{\beta + 2 \delta} \]  |
| \( r \)\\ Intrinsic growth rate  |  \[  \frac{\alpha \beta}{\beta+\delta} \]  |  \[  \frac{\alpha \beta}{2(\beta + 2 \delta)} \]  |
| \( K \)\\ Carrying capacity  |  \[  \frac{\alpha \beta - (\alpha + \gamma) \delta }{(\beta + \delta) \delta} \]  |  \[  \frac{\alpha \beta - 4 (\alpha + \gamma) \delta }{2 (\beta + 2 \delta) \delta} \]  |

Numerical simulations show both agree well with the analytic QSSA predictions, even when the QSSA is not expected to apply.